vault backup: 2026-10-05 11:54:11

College/AU 26/MATH 224 (Multivar)/Class 10-5.md

@@ -12,3 +12,7 @@ This can be extended to more general domains that are not rectangles such as $[a
 Example:
 finding the volume under the function $f(x,y) = xy$ over the domain bound by 
 $$\begin{gathered} 1<x<2 \\[1.5ex] x < y < x^2 \end{gathered}$$
+We can think of finding the volume of the solid by using slices parallel to the $YZ$ plane. With the domain provided this area can be found with 
+$$ Area(x^*)=\int_{x}^{x^2}x^*ydy$$
+Since $x$ is bound by $1<x<2$ the double integral to find the volume can be written as 
+$$\begin{gather} \int_{1}^2 \left[ \int_{x}^{x^2}xy \  dy \right] dx \\[1.5ex] \int_{1}^2 \left[ \int_{x}^{x^2} \frac{xy^2}{2}   \right] dx \end{gather}$$